梁—质量块系统的动力学行为研究
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摘要
随着土木和机械工程的迅速发展,梁-质量块(包括轴向移动和固定)系统的振动与稳定性研究显得尤为重要。本文分别基于Bernoulli-Euler梁理论和Timoshenko梁理论,系统地研究了移动质量块作用下梁结构的振动特性、非线性动力学与稳定性机理。研究内容包括如下几个方面:单个或多个移动质量块作用下Bernoulli-Euler梁的横向振动、弯扭耦合振动和非线性动力学响应,以及Timoshenko梁的横向振动。特别是计入了质量块的惯性效应,目的是为工程梁结构的振动分析建立更有效的动力学方程和提供更实际的研究思路。
在研究Bernoulli-Euler梁结构的线性横向振动时,考虑单个移动质量块的惯性效应和集中阻尼器的影响,建立了两端可变约束下梁结构的一般性振动方程,用来计算多种边界条件下梁的固有频率。研究表明,质量块轴向移动速度、加速度和质量比对梁频率影响较大;另外,当梁结构两端约束不对称时,在某些轴向移动速度下梁的振动频率可能发生突变。
研究了单个移动质量块以及轴向谐激励作用下的Bernoulli-Euler梁结构的非线性横向强迫振动。考虑质量块的惯性效应,首先导出具有5次非线性项的运动方程;然后基于Galerkin法进行离散,通过大量数值算例详细探讨了Galerkin法模态截断数对系统动力响应的影响,数值计算结果表明:当计及高阶模态时,梁的时程曲线可能叠加无序挠动。在以上研究基础上,运用多尺度法求解了单模态截断下的动力学方程,分析了系统在1/2亚谐共振-主参数共振情况下的稳定性和局部分岔,研究结果表明:在系统不同的参数区域内,简支柔性梁可能出现不同的横向振动形式。
在工程实际中,梁的弯曲中心线与扭转中心线可能不重合。本文研究了含移动质量块Bernoulli-Euler梁结构弯曲/扭转耦合的线性振动。考虑质量块的惯性效应,建立单个质量块作用下梁结构弯扭耦合振动的控制方程,详细研究了梁在外加扭力矩作用下的动力学行为。数值计算表明,梁质量和质量块轴向移动速度对梁的动力响应影响较大;另外,梁的转动惯量对梁的自振频率也有较大影响。
研究了多车道Bernoulli-Euler梁-多个来往质量块系统的线性横向振动。计入质量块的惯性效应,导出系统的偏微分振动方程,该方程经Galerkin离散后形成系统的动力学方程组。数值计算中,重点检测了来往质量块情形下多车道梁结构的运动形态,以及各参数对系统动力响应的影响。这一工作为多车道高架桥梁问题的进一步研究提供了基础。
基于Timoshenko梁理论,考虑剪切变形和转动惯量的影响,计及移动质量块的惯性效应,建立了Timoshenko梁结构横向振动的控制方程;在此基础上,研究了梁结构的动力学行为以及Galerkin法模态截断数的影响。数值结果表明,对于细长梁,Bernoulli-Euler梁理论和Timoshenko梁理论所预测的结构动力学行为比较一致;对于短粗梁,由于剪切变形和转动惯量对系统动力学行为影响较大,Timoshenko梁理论所预测的梁振动频率比前者的小。
With the rapid development of civil engineering and mechanical engineering, it is important to study the vibration and stability of a beam carrying masses system, for both concentrated and axial moving masses. Based on Bernoulli-Euler beam theory and Timoshenko beam theory, the vibration characteristics, stability and nonlinear dynamics of a beam carrying masses are investigated respectively. Research include following sections: the transverse vibration, bending-torsional coupling vibration and nonlinear dynamic response of Bernoulli-Euler beam carrying single or multiple moving mass; the transverse vibration of Timoshenko beam carrying single moving mass. Considering the inertia effect of the masses, this work establishes effective dynamic equation and supplies as a reliable theory base for the analysis in engineering.
Considering the inertia effect of single moving mass and concentrated dashpot, the linear transverse vibration of Bernoulli-Euler beam is studied and a general equation of the motion is derived under flexible constraints. Based on the motion equation, the natural frequencies of the beam with various boundary conditions are calculated. It shows that the frequencies of beam are affected by axial moving velocity, acceleration of single mass and mass ratio. Besides, under asymmetric constraints, certain axial velocity of moving mass leads to rapid change of beam frequency.
Subjected to single moving mass and axial harmonic excitation, the nonlinear transverse vibration of Bernoulli-Euler beam is investigated. Considering the inertia effect of moving mass, the motion equation with the fifth-order nonlinear term is derived. Based on Galerkin method, the effect of Galerkin modal truncation on the dynamic behavior for the system is studied by various numerical examples. Numerical results show that, the disorder may occur at high order modes. For single mode model, the dynamic equation is solved by multi-scales method. In addition, the stability and local bifurcation of the system are analyzed for 1/2 sub harmonic resonance, which indicate simply-supported flexible beam has different transverse vibration in various system parameter regions.
In engineering, the misalignment of mass centers and the shear centers may exist. The linear bending-torsional coupling vibration of Bernoulli-Euler beam carrying single moving mass is studied. Considering the inertia effect of single moving mass, the governing equations of bending-torsional coupling vibration are established. Meanwhile, the dynamic behavior of the beam by external torque is discussed. Numerical analysis shows, the mass per unit length of beam and the axial velocity of moving mass have significant influence on beam vibration; the frequencies of the beam are greatly affected by the moment of inertia.
Transverse vibration of multi-channel beam carrying several toing-and-froing masses is analyzed. Considering the inertial effect of masses, the partial differential vibration equation of system is derived and simplified by using mode analysis and Galerkin integral. In numerical simulation examples, the motion forms of the beam are detected, the effect on the dynamic response of the system is investigated by various parameters as well. Furthermore, this work provides a theory base for multiple-lane high-speed bridge problem.
According to Timoshenko beam theory, the governing equation of transverse vibration of the beam including shear deformation and rotator inertia is obtained. Considering the inertia effect of single moving mass, the effect of Galerkin modal truncation on the dynamic behavior of beam structure is investigated.
For slender beam systems, quite similar behavior was observed in Bernoulli-Euler beam and Timoshenko beam theory. For short and wide beam, because of large effect of shear deformation and rotary inertia on dynamic behavior of beam, Timoshenko beam theory predicts smaller beam vibration frequencies than Bernoulli-Euler beam theory does.
Comparing with Bernoulli-Euler beam, quite similar behavior was observed on these two slender beam systems. For short and wide beam, because of the large effect on shear deformation and rotary inertia of short and wide beam, the vibration frequencies of beam, which predict by Timoshenko beam theory, are smaller than the former.
在研究Bernoulli-Euler梁结构的线性横向振动时,考虑单个移动质量块的惯性效应和集中阻尼器的影响,建立了两端可变约束下梁结构的一般性振动方程,用来计算多种边界条件下梁的固有频率。研究表明,质量块轴向移动速度、加速度和质量比对梁频率影响较大;另外,当梁结构两端约束不对称时,在某些轴向移动速度下梁的振动频率可能发生突变。
研究了单个移动质量块以及轴向谐激励作用下的Bernoulli-Euler梁结构的非线性横向强迫振动。考虑质量块的惯性效应,首先导出具有5次非线性项的运动方程;然后基于Galerkin法进行离散,通过大量数值算例详细探讨了Galerkin法模态截断数对系统动力响应的影响,数值计算结果表明:当计及高阶模态时,梁的时程曲线可能叠加无序挠动。在以上研究基础上,运用多尺度法求解了单模态截断下的动力学方程,分析了系统在1/2亚谐共振-主参数共振情况下的稳定性和局部分岔,研究结果表明:在系统不同的参数区域内,简支柔性梁可能出现不同的横向振动形式。
在工程实际中,梁的弯曲中心线与扭转中心线可能不重合。本文研究了含移动质量块Bernoulli-Euler梁结构弯曲/扭转耦合的线性振动。考虑质量块的惯性效应,建立单个质量块作用下梁结构弯扭耦合振动的控制方程,详细研究了梁在外加扭力矩作用下的动力学行为。数值计算表明,梁质量和质量块轴向移动速度对梁的动力响应影响较大;另外,梁的转动惯量对梁的自振频率也有较大影响。
研究了多车道Bernoulli-Euler梁-多个来往质量块系统的线性横向振动。计入质量块的惯性效应,导出系统的偏微分振动方程,该方程经Galerkin离散后形成系统的动力学方程组。数值计算中,重点检测了来往质量块情形下多车道梁结构的运动形态,以及各参数对系统动力响应的影响。这一工作为多车道高架桥梁问题的进一步研究提供了基础。
基于Timoshenko梁理论,考虑剪切变形和转动惯量的影响,计及移动质量块的惯性效应,建立了Timoshenko梁结构横向振动的控制方程;在此基础上,研究了梁结构的动力学行为以及Galerkin法模态截断数的影响。数值结果表明,对于细长梁,Bernoulli-Euler梁理论和Timoshenko梁理论所预测的结构动力学行为比较一致;对于短粗梁,由于剪切变形和转动惯量对系统动力学行为影响较大,Timoshenko梁理论所预测的梁振动频率比前者的小。
With the rapid development of civil engineering and mechanical engineering, it is important to study the vibration and stability of a beam carrying masses system, for both concentrated and axial moving masses. Based on Bernoulli-Euler beam theory and Timoshenko beam theory, the vibration characteristics, stability and nonlinear dynamics of a beam carrying masses are investigated respectively. Research include following sections: the transverse vibration, bending-torsional coupling vibration and nonlinear dynamic response of Bernoulli-Euler beam carrying single or multiple moving mass; the transverse vibration of Timoshenko beam carrying single moving mass. Considering the inertia effect of the masses, this work establishes effective dynamic equation and supplies as a reliable theory base for the analysis in engineering.
Considering the inertia effect of single moving mass and concentrated dashpot, the linear transverse vibration of Bernoulli-Euler beam is studied and a general equation of the motion is derived under flexible constraints. Based on the motion equation, the natural frequencies of the beam with various boundary conditions are calculated. It shows that the frequencies of beam are affected by axial moving velocity, acceleration of single mass and mass ratio. Besides, under asymmetric constraints, certain axial velocity of moving mass leads to rapid change of beam frequency.
Subjected to single moving mass and axial harmonic excitation, the nonlinear transverse vibration of Bernoulli-Euler beam is investigated. Considering the inertia effect of moving mass, the motion equation with the fifth-order nonlinear term is derived. Based on Galerkin method, the effect of Galerkin modal truncation on the dynamic behavior for the system is studied by various numerical examples. Numerical results show that, the disorder may occur at high order modes. For single mode model, the dynamic equation is solved by multi-scales method. In addition, the stability and local bifurcation of the system are analyzed for 1/2 sub harmonic resonance, which indicate simply-supported flexible beam has different transverse vibration in various system parameter regions.
In engineering, the misalignment of mass centers and the shear centers may exist. The linear bending-torsional coupling vibration of Bernoulli-Euler beam carrying single moving mass is studied. Considering the inertia effect of single moving mass, the governing equations of bending-torsional coupling vibration are established. Meanwhile, the dynamic behavior of the beam by external torque is discussed. Numerical analysis shows, the mass per unit length of beam and the axial velocity of moving mass have significant influence on beam vibration; the frequencies of the beam are greatly affected by the moment of inertia.
Transverse vibration of multi-channel beam carrying several toing-and-froing masses is analyzed. Considering the inertial effect of masses, the partial differential vibration equation of system is derived and simplified by using mode analysis and Galerkin integral. In numerical simulation examples, the motion forms of the beam are detected, the effect on the dynamic response of the system is investigated by various parameters as well. Furthermore, this work provides a theory base for multiple-lane high-speed bridge problem.
According to Timoshenko beam theory, the governing equation of transverse vibration of the beam including shear deformation and rotator inertia is obtained. Considering the inertia effect of single moving mass, the effect of Galerkin modal truncation on the dynamic behavior of beam structure is investigated.
For slender beam systems, quite similar behavior was observed in Bernoulli-Euler beam and Timoshenko beam theory. For short and wide beam, because of large effect of shear deformation and rotary inertia on dynamic behavior of beam, Timoshenko beam theory predicts smaller beam vibration frequencies than Bernoulli-Euler beam theory does.
Comparing with Bernoulli-Euler beam, quite similar behavior was observed on these two slender beam systems. For short and wide beam, because of the large effect on shear deformation and rotary inertia of short and wide beam, the vibration frequencies of beam, which predict by Timoshenko beam theory, are smaller than the former.
引文
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